MIT OpenCourseWare
http://ocw.mit.edu
18.727 Topics in Algebraic Geometry: Algebraic Surfaces
Spring 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
ALGEBRAIC SURFACES, LECTURE 15
LECTURES: ABHINAV KUMAR
1. Elliptic Surfaces (contd.)
Assume we are over C. Given f : X → C, we can associate the functional in­
variant j : C → P1 , c �→ j(fc ) and a homological invariant: let c1 , . . . , cn be points
over which the fibers are singular, C ∗ = C � {c1 , . . . , cn } and X ∗ = f −1 (C ∗ );
then we have the sheaf R1 f∗ Z, which is a homological invariant; (R1 f∗ Z) ⊗C C =
H 1 (FC , Z), so it is a locally constant sheaf. We get a representation G : π1 (C ∗ ) →
SL2 (Z) called the global monodromy. The equivalence class of this representation
is called the homological invariant. The local monodromy at a point ci is the
image of a loop around that point in SL2 (Z). We can write down the �conjugacy
�
1 n
class of the local monodromy for the bad fibers: for a In fiber, it is
.
0 1
J and G determine the elliptic fibration up to isomorphism. Given any elliptic
fibration, we get a corresponding Jacobian fibration which does have a section
O : C → X.
Let X → C be an elliptic fibration with section
�), and let the class of the
fiber be F . For each singular fiber, let Fv = Θv,0 + Mv,i Θv,i , where Θv,0 is the
identity component. Algebraic and numerical equivalence are the same for elliptic
surfaces, and KX = f ∗ (KC − N ), where N is isomorphic to the normal bundle of
the zero section O in X. Note that OC (N ) = R1 f∗ OX . Thus, KS ≡ (2g−2+χ)F ,
where χ = the “arithmetic genus” of X = h0 (OX ) − h1 (OX ) + h2 (OX ), with
(KS )2 = 0 and deg N = 02 = −χ. For any section P we have
(1)
2g − 2 = P (P + K) = P 2 + P · K = P 2 + 2g − 2 + χ
so P 2 = −χ.
Let T be the subgroup of N S(X) generated by the zero section O and all
irreducible components of fibers (called the trivial lattice)
(2)
X(C) ∼
= E(K) ∼
= N S(X)/T
where K = k(C). Here T is torsion free, and the map is given by P ∈
�E(K) �→ P
mod T . We can obtain the Shioda-Tate formula: rk N S(X) = 2 + (mv − 1) +
rk , E(k), where rk E(k) is the Mordell-Weil rank.
1
2
LECTURES: ABHINAV KUMAR
Remark. An open arithmetic question: is the rank of E(K) unbounded for K =
K(C), the function field of a curve (e.g. C = P1 , E(Q(x)))? For Fp (k), the
answer is yes (Shafarevich-Tate). For E(Q), we can write down a finite list of
the possible torsion (Mazur’s theorem).
2. Kodaira dimension 0
If X is a surface with κ(X) = 0, then (K 2 ) = 0, pg ≤ 1 (since K 2 < 0
would imply K ruled): we will show this later. Noether’s formula says that
10 − 8q + 12pg = b2 + 2Δ, where Δ = 2(q − s). Then 0 ≤ Δ ≤ 2pg and Δ even
implies that it is 0 or 2. Now, the four possibilities based on combinations of
(b1 , b2 ) are:
(0, 10) are Enriques surfaces, “classical” if (pg , q, Δ) = (0, 0, 0) and “non-classical”
if (pg , q, Δ) = (1, 1, 2).
(2, 2) are bielliptic or hyperelliptic surfaces if (pg , q, Δ) = (0, 1, 0), and quasi­
hyperelliptic surfaces if (pg , q, Δ) = (1, 2, 2).
(0, 22) are K3 surfaces, with χ(OX ) = 0, q = 0, pg = 1, Δ = 0, and
(4, 6) are abelian surfaces, with χ(OX ) = 0, q = 2, pg = 1, Δ = 0.
Note that, by the above formulae, there is another possible combination: (b1 , b2 ) =
(2, 14), q1 , pg = 1, Δ = 0.
Proposition 1. No surface exists with this combination of invariants.
Proof. Since b1 = 2 = 2s, the Picard variety of X has dimension 1. So ∃
a line bundle L0 on X difference from OX but algebraically equivalent to 0.
Since K ≡ 0 and pg = 1, we get KX ∼ OX . Applying Riemann-Roch to L
gives χ(L) = 12 L · (L ⊗ K −1 ) + χ(OX ) = χ(OX ) = 1. Thus, H 0 (L > 0 or
�
H 0 (L−1 ) = H 2 (L) > 0. But L ≡ 0 gives L ∼ OX , a contradiction.
2.1. Abelian surfaces. These are smooth, complete group varieties of dimen­
sion 2 over k. For the general theory, we refer to Mumford’s Abelian Varieties
and Birkenhaake-Lange’s Complex Abelian Varieties. Over C, these are com­
plex tori: C2 /Λ, equipped with a Riemann form ψ : Λ × Λ → Z. This form is
alternating, bilinear, and satisfies the following two conditions:
• ψR : C2 × C2 → R satisfies ψR (iv, iw) = ψR (v, w) for v, w ∈ C × C.
• The associated Hermitian form H(v, w) = ψR (iv, w)+iψR (v, w) is positive
definite.
This gives rise to a line bundle L on C2 /Λ which is the class of an ample line
bundle. L is called a polarization of the abelian variety, and
�
(3)
h0 (L) = disc (φ) = |Pf (ψ)|
In general, over any characteristic, A∨ = Pic 0 A is the dual abelian variety, and
it is automatically reduced. A polarization is an isogeny A → A∨ : for example,
ALGEBRAIC SURFACES, LECTURE 15
3
whenever we have an ample line bundle L on A, the subgroup K(L) = {x ∈
A | Tx∗ L ∼
= L} is finite. The map A → A∨ , x �→ Tx∗ L ⊗ L−1 gives an isogeny with
finite kernel K(L), i.e. a polarization. For a line bundle L,
(4)
χ(L) =
Lg
L2
=
, χ(L)2 = deg φL
dim A
2
The polarization is principal if it has degree 1. There are two common examples
of abelian surfaces with principal polarization:
(1) J(C), C a genus 2 curve. Here, we can explicitly write down equations for
J(C) (e.g. Cassels-Flynn, Prolegomena to a Middlebrow Arithmetic of
Curves of Genus 2). The image of C is J(C) under x �→ [x] − [x0 ], where
x0 ∈ C is a fixed point, is the theta divisor and it generates N S(J(C))
for a generic C. We have C 2 = C · (C + K) = 2g − 2 = 2 since K ≡ 0
for an abelian variety. The theta divisor gives the principal polarization
2
χ(OX (C)) = C2 = 1, so deg φC = χ(OX (C))2 = 1.
(2) Fix E2 with E1 , E2 elliptic curves. Then E1 + E2 gives a principally
polarized abelian variety, with E1 ·E2 = 1, Ei2 = 0 =⇒ (E1 +E2 )2 /2 = 1.
The moduli space of principally polarized abelian varieties is made up of these
two types of points, with the second type forming the boundary.
There are lots of arithmetic questions here. By the Mordell-Weil theorem, if
A is defined over a number field K, then A(K) is finitely generated for a number
field L containing K. What is the structure of the torsion subgroup, and what
can you say about the rank? Merel showed that, given a positive integer d, there
is a constant Bd , depending only on d, s.t. for any number field K of degree
d over Q, and any elliptic curve E/K with a point of order n, n ≤ Bd . This
also bounds the size of E(K)tors . This is called the strong uniform boundedness
conjecture. This is a broad generalization of Mazur and Kamienny’s work. The
analogue for abelian surfaces is not known. Also, we don’t know much about
ranks. The Galois representations that arise are of course very important. For
instance, considering the action of Gal (K/K) on A[N ] for N prime gives a map
to the symplectic group Sp 4 (FN ).
2
∼ 6
For an abelian surface,
Hét
�
�(A, Q� ) = Q� with symplectic form induced from
0 1
H 2 (A, Z) ∼
. N S has rank between 1 and 4 over C (because of
= U 3, U =
1 0
Hodge theory), e.g. A = J(C) (rk NS = 1, C generic), A = E1 × E2 (rk NS = 2,
E1 not isogenous to E2 ), A = E1 × E1 (rk NS = 3, E1 not CM), and A = E1 × E1
(rk NS = 4, E1 CM). In characteristic p, we can also have rank 6 for E1 × E1 ,
E1 a supersingular elliptic curve.
2.2. K3 surfaces. These have b2 = 22, b1 = 0, χ(OX ) = 2, q = 0, pg = 1, Δ = 0.
This is equivalent to saying KX ∼
= OX and q = 0. (exercise).
4
LECTURES: ABHINAV KUMAR
Proposition 2. Pic τ (X) = 0, i.e. Pic (X) is torsion free, i.e. numerical,
algebraic, and linear equivalence are the same, implying that Pic (X) is a free
abelian group of finite rank. Moreover, if f : X � → X is a connected étale cover
of X, then f is an isomorphism, i.e. X is algebraically simply connected.
Proof. If L is s.t. L ≡ 0 then, by Riemann-Roch, we get (since K ∼ 0 =⇒ K ≡
0)
1
(5)
χ(L) = χ(OX ) + L · (L + K) = χ(OX ) = 2 =⇒ h0 (L) �= 0
2
� 0: since L ≡ 0, we must have L ∼
or H 2 (L) = H 0 (L−1 )) =
= OX or
L−1 ∼
(6)
= OX =⇒ L ∼
= OX =⇒ Pic τ = 0
[let D ∈ H 0 (L): if D were effective then L · H = D · H > 0 for H ample]. For
the second part, let f : X � → X be étale of degree n. ωX � = f ∗ (ωX ) ∼
= OX � from
(7)
0 → f ∗ (Ω1X/k ) → Ω2X/k → Ω1X � /X = 0
. So κ(X � ) = 0, X � is a minimal surface, and pg (X � ) = 1 =⇒ X � is in in the list
of possibilities for Kodaira dimension 0. Also χ(OX � ) = nχ(OX ) = 2n. But no
�
surface in the list has χ(OX � ) > 2 =⇒ n = 1.